Step 1: Prepare the Graham Cracker Crust

1. Preheat the oven to 350°F (175°C). Grease a 9-inch springform pan with butter or non-stick spray.
2. In a medium bowl, combine the graham cracker crumbs, sugar, and melted butter. Mix until the crumbs are evenly coated.
3. Press the crumb mixture into the bottom of the prepared pan, creating an even layer. Use the back of a spoon or the bottom of a measuring cup to press firmly.
4. Bake the crust for 8-10 minutes until it is golden brown. Remove from the oven and let it cool while you prepare the filling.

Step 2: Prepare the Cheesecake Filling

1. In a large bowl, beat the cream cheese and sugar with an electric mixer on medium speed until smooth and creamy, about 2-3 minutes.
2. Add the eggs one at a time, beating well after each addition. Scrape down the sides of the bowl as needed to ensure everything is fully incorporated.
3. Add the sour cream and vanilla extract, mixing just until combined.
4. Pour half of the filling into the prepared crust. Drizzle the blackberry puree over the filling and swirl it gently with a knife to create a marbled effect. Pour the remaining cheesecake filling on top, smoothing the surface with a spatula.

Step 3: Bake the Cheesecake

1. Place the springform pan in a large roasting pan. Add enough hot water to the roasting pan to come halfway up the sides of the springform pan. This water bath method helps the cheesecake cook evenly and prevents cracks.
2. Bake the cheesecake for 55-65 minutes, or until the edges are set but the center is still slightly jiggly.
3. Turn off the oven and crack the door open. Let the cheesecake cool in the oven for 1 hour to prevent cracks from forming.
4. Remove the cheesecake from the oven and water bath, and let it cool to room temperature on a wire rack. Refrigerate for at least 4 hours, preferably overnight, to fully set.

Step 4: Prepare the Blackberry Topping

1. In a medium saucepan, combine the blackberries, sugar, and lemon juice. Cook over medium heat, stirring occasionally, until the blackberries release their juices and begin to break down, about 5-7 minutes.
2. Stir in the cornstarch slurry (cornstarch mixed with water) and continue to cook, stirring constantly, until the mixture thickens, about 2-3 minutes.
3. Remove the sauce from the heat and let it cool to room temperature before spreading it over the chilled cheesecake.

Step 5: Serve the Cheesecake

1. Once the cheesecake is fully chilled, remove it from the springform pan.
2. Spread the blackberry topping evenly over the top of the cheesecake.
3. Garnish with fresh blackberries, whipped cream, and mint leaves, if desired.
4. Slice and serve. Enjoy!